Some of you are already insisting in your head that 6 ÷ 2(1+2) has only one right answer, but hear me out. The problem isn’t the mathematical operations. It’s knowing what operations the author of the problem wants you to do, and in what order. Simple, right? We use an “order of operations” rule we memorized in childhood: “Please excuse my dear Aunt Sally,” or PEMDAS, which stands for Parentheses Exponents Multiplication Division Addition Subtraction.* This handy acronym should settle any debate—except it doesn’t, because it’s not a rule at all. It’s a convention, a customary way of doing things we’ve developed only recently, and like other customs, it has evolved over time. (And even math teachers argue over order of operations.)
“In earlier times, the conventions didn’t seem as rigid and people were supposed to just figure it out if they were mathematically competent,” says Judy Grabiner, a historian of mathematics at Pitzer College in Claremont, Calif. Mathematicians generally began their written work with a list of the conventions they were using, but the rise of mass math education and the textbook industry, as well as the subsequent development of computer programming languages, required something more codified. That codification occurred somewhere around the turn of the last century. The first reference to PEMDAS is hard to pin down. Even a short list of what different early algebra texts taught reveals how inconsistently the order of operations was applied.
So that brings us back to 6 ÷ 2(1+2). There are three ways to think about this problem—and none is incorrect. (If you don’t believe me, plug it into a few different calculators, or even check out Google, where commenters have argued over Google’s calculator answer.)
One way is to interpret the obelus, or ÷ symbol, as dividing everything to the left of it by everything to the right of it. Textbooks don’t typically use the symbol that way today, but it has been used that way historically. If you calculate the problem using this convention, it’s 6 divided by (2(1+2)), which is 1. Typically, though, if the author wanted you to interpret it that way, she would have used parentheses to indicate as much.
You can alternatively apply PEMDAS as schools do today: Simplify everything inside the parentheses first, then exponents, then all multiplication and division from left to right in the order both operations appear, then all addition and subtraction from left to right in the order both operations appear. (A better acronym would be PEMA, actually, to make it clear that multiplication and division are done together, and addition and subtraction are done together.) By that convention, 6 ÷ 2(1+2) = 6 ÷ 2 × (1+2) = 6 ÷ 2 × 3 = 3 × 3 = 9. If you were taking the ACT, SAT, or GRE (which would probably use parentheses to eliminate confusion), this method would yield the correct answer.
But wait, you say—isn’t that 2 to the left of the parentheses part of simplifying the parentheses? After all, this is what my own Facebook debate partners were arguing. In fact, the 2 is not part of the “P” in PEMDAS for simplifying parentheses, but there is a basis for simplifying the 2(2+1) before it’s divided by 6. It’s called “implied multiplication by juxtaposition.” We know the expression 5x means to multiply 5 and x because they are juxtaposed next to one other. But should these operations be done before a division that occurs to the left of them in a problem? That depends on whom you’re talking to, or what calculator or programming language you’re using.
Internet rumors claim the American Mathematical Society has written “multiplication indicated by juxtaposition is carried out before division,” but no original AMS source exists online anymore (if it ever did). Still, some early math textbooks also taught students to do all multiplications and then all divisions, but most, such as this 1907 high-school algebra textbook, this 1910 textbook, and this 1912 textbook, recommended performing all multiplications and divisions in the order they appear first, followed by additions and subtractions. (This convention makes sense as well with the Canadian and British versions of PEMDAS, such as BEDMAS, BIDMAS, and BODMAS, which all list division before multiplication in the acronym.) The most sensible advice, in a 1917 edition of Mathematical Gazette, recommended using parentheses to avoid ambiguity. (Duh!) But even noted math historian Florian Cajori wrote in A History of Mathematical Notations in 1928-29, "If an arithmetical or algebraical term contains ÷ and ×, there is at present no agreement as to which sign shall be used first."
If it “feels” natural to you that implied multiplication takes precedence over division (whether because it’s next to a parentheses or not), then you would get 6 ÷ 2(1+2) = 6 ÷ (2(3)) = 6 ÷ 6 = 1. That answer would be incorrect on most U.S. standardized tests, but you wouldn’t necessarily be wrong. (Insert rant against standardized tests here.) You would just be in the minority about which convention you’re using.
Still unconvinced that arguing over math problems is similar to arguing over whether to use a plural or singular pronoun with indefinite pronouns? Let’s return to the obelus (÷) because a brief history of division signs reveals the ambiguity of the syntax of math. Nearly a half-dozen division signs have been recorded in mathematical notation. The colon was used in a 1633 text, which seems odd until you realize we still use it in ratios (2:3 is commonly the same as 2/3 in ratios).
Even before that, a close parentheses was used in the 1540s, so that 8)24 meant 24 ÷ 8. Again, that looks odd, but we still use it today in long division. It just looks different because we combine it with a different symbol, the lengthy vinculum (--------) across the top, to group together the numbers to be divided. The vinculum is also used over repeating decimal digits and with radicals (√ is used with -------- across the top); you probably just didn’t realize the square root sign was a mashup of two math symbols. A vinculum usually has little to do with division; it’s used in fractions and to group together numbers just as parentheses are.
You might expect 10 ÷ 5 is the same as 10/5 is the same as 10 over a 5 with a vinculum between them, but each has its own eccentricities. We’ve already noted that ÷ can mean “divide the number on the left by the number on the right” or “divide the expression on the left by the expression on the right.” But it gets really tricky when people assume that a slash replaces a vinculum. Does ab/cd = (ab)÷(cd) or ((ab)÷c)÷d? Does a/b/c mean (a)÷(b)÷(c) or a÷(b/c) or (a/b)÷c? (Answer: Use some parentheses!)
The bottom line is that “order of operations” conventions are not universal truths in the same way that the sum of 2 and 2 is always 4. Conventions evolve throughout history in response to cultural and technological shifts. Meanwhile, those ranting online about gaps in U.S. math education and about the “right” answer to these intentionally ambiguous math problems might be, ironically, missing a bigger point.
“To my mind,” says Grabiner, “the major deficit in U.S. math education is that people think math is about calculation and formulas and getting the one right answer, rather than being about exciting ideas that cut across all sorts of intellectual categories, clear and logical thinking, the power of abstraction and a language that lets you solve problems you’ve never seen before.” Even if that language, like any other, can be a bit ambiguous sometimes.
Correction, March 12, 2013: This article originally misstated the mnemonic for the PEMDAS order of operations rule. It's "excuse," not "forgive." (Return to the corrected sentence.)